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Full text of "Treatise On Analysis Vol-Ii"

14    INTEGRATION WITH RESPECT TO A POSITIVE MEASURE       189

0 and a mass — n at the point l/Ğ. Then the sequence (//,„( A)) is bounded for every open
set A which is |ftn|-quadrable for all n, but the sequence of norms \\fj,n\\ is unbounded.
Again, if we take ^n to be the measure defined by a mass n at the point l/n and a mass
— n at the point l/(n -f- 1), then the sequence (ftM(A)) is bounded for every interval
A c [0, 1 ] (and therefore also for every finite subset A of [0, 1 ]), but the sequence of
norms ||/xn|! is not bounded.

2. For each integer n J> 1, let En be an at most denumerable closed subset of I = ]0, 1[,
and let */„ be the family of component intervals of I — En . Suppose that the maxi-
mum length dn of the intervals J e Jn tends to 0 as* n -*• -f oo . Let A be Lebesgue
measure on I and let A be a A-measurable subset of I. Suppose that there exists a number
k e ]0, 1 [ such that, for all n and all J e Jn , we have A(A n J) < AA(J). Show that
A(A) = 0. (By using (13.7.9), show that A(A) ^ e + &(A(A) + e) for all s > 0.)

3. For each integer n > 0, the Farey series of order n is the set Fn of all rational numbers
which, when expressed in their lowest terms plq, are such that Q^p^q^n, and
arranged in increasing order. The distance between two consecutive terms of Fn is
therefore ^l/n.

(a)    Show  that  if two  rational   numbers  r~p/g  and  r'=p'/q'  are such that
VP' — Ptf = ħ 1> tnen f°r every pair of integers (p"9 #") there exist integers jc, y such that
p" — px + p'y and q" — qx + q'y. The fraction p*l<f belongs to the closed interval with
endpoints r, r' if and only if x, ^ are of the same sign.

(b)    Deduce from (a) that if r=*p/q and r' — p'lq' are two rational numbers belonging
to the interval [0, 1], such that q > 0, q' > 0 and qp' — pq' == ħ 1, then r and rf are
consecutive elements of the Farey series Fsup(9, fl/> . Moreover the smallest of the integers
w such that the open interval with endpoints r, r' contains a point of Fm is the integer
q -h <?', and this open interval contains only one point of Fq+q,, namely (p +/>')/(<? + #')•

(c)    Conversely, if r and r' are consecutive terms of Fn, show that qp' —pqf = ħ1
(induction on /?).

4. For each x e I = ]0, 1], put /o(;t) = (1/x) — [l/#], where [r] denotes the integral part
of the real number /. lfp"(x) = p(p"""1(x)) is defined and nonzero, put qn + i(x) = [l//3n(x)]

(a)   For every Ğ such that p"(:c) is defined, show that

An- !(*>*(*) -fAn(*)

where An(jc) and Bn(jc) are integers >/i — 1. The fractions An_i(Ar)/Bn_i(A:) and
An(^:)/Bn(x) are two consecutive terms of a Farey series (Problem 3) and x belongs to
the closed interval with these as endpoints. Deduce that pn(x) is defined for all/? ;> 1 if
and only if x is irrational.

The (finite or infinite) sequence of numbers qn(x) is called the continued fraction
expansion of x, and the fractions An(x)/Bn(x) are the convergents to x. The An(x) and
Bn(x) are constant on the complement of the denumerable closed set En of points x
such that pn+l(x) is not defined, and on each of the component intervals of I — En
the function pn is monotone and varies from 0 to 1 .

(b) Let A be Lebesgue measure on I and let A be a A-measurable subset of I consisting
entirely of irrational numbers. Suppose that <pA is X-invariant with respect to p (SectionG in E x R.